Problem
ALG-B1-M06-P013 A system with one hidden quantity
#13
★★★☆☆ Level 3 of 5
Find all real pairs \((x,y)\) such that \(x+y=3\) and \(x^2+y^2+xy=7\).
Express \(x^2+y^2+xy\) through \(s=x+y\) and \(p=xy\).
Let \(s=x+y=3\), \(p=xy\). Then \(x^2+y^2+xy=s^2-p\). We get \(9-p=7\), hence \(p=2\).
The numbers \(x,y\) are roots of \(t^2-3t+2=0\), so \(\{x,y\}=\{1,2\}\). Answer: \((1,2)\), \((2,1)\).
The solution is short, but it requires the right hidden substitution.